Squeezing flow of viscoplastic fluids subject to wall slip

Polymer processing operations such as compression molding, sheet forming and injection molding can be modeled by squeezing flows between two approaching parallel surfaces in relative motion. Squeezing flows also find applications in the modeling of lubrication systems, and in the determination of rheological properties. Here, analytical solutions are developed for the constant-speed squeezing flow of viscoplastic fluids. It is assumed that the fluid is purely viscous, and hence viscoelastic effects unimportant. The rheological behavior of the viscoplastic fluids is represented by the Herschel-Bulkley viscosity function. The deformation behavior of commonly encountered viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which is a function of the wall shear stress. The slip coefficient that relates the slip velocity to the shear stress is affected by the material of construction and also the roughness of the solid surfaces, leading to the possibility of different slip coefficients at various solid surfaces. The model developed in this study accommodates the use of different slip coefficients at different solid surfaces. The accuracy of the solutions is established, and the effects of various parameters such as slip coefficient and apparent yield stress are examined. The solutions provide useful design expressions that can be utilized for squeezing flows of viscoplastic fluids, with or without wall slip at the solid boundaries.     

Modelling of viscoelastic coextrusion flows in multi-manifold flat dies

Abstract

A recently proposed modification of the viscoelastic Leonov model is employed as a stress calculator in FEM analysis with a full u-v-p-t numerical scheme for coextrusion flow in multimanifold flat dies with 30° and 90° entrance angles. It is shown that the predicted stresses, interface location and streamline fields are in good agreement with the measurements. It is also shown that extensional viscosity has to be used in the modelling of the coextrusion flow to confirm experimental data.     

The effect of wall slip on the performance of fiat extrusion dies

Flat extrusion dies are commonly used in a wide variety of film. Sheet and coating applications. Although flal dies can be designed to produce an exit flow distribution that is very uniform across most of the width, there will usually be a region along each side where it drops gradually to zero. This often requires trimming the edges of the film or sheet downstream in order to meet product specifications. It is commonly believed that treating the land area of the die with coatings that promote a small amount of wall slip will reduce the size of this edge effect and therefore improve die performance. This analysis shows that slip over the entire land region of the die will adversely affect die performance. Better performance is possible but only if the sides of the land are treated.     

Mechanics of steady flow in coextrusion fiber spinning

The steady flow of isothermal bicomponent coextrusion fiber spinning has been investigated. A model has been chosen in which a Newtonian fluid and a Phan-Thien/Tanner (PTT) fluid were considered to be the core and the skin layer, respectively. This model was adopted to study the effect on spinline velocity of an interaction between two fluids with quite different extensional rheology. The effects of the hoop stress, gravity, inertia, and surface and interfacial tensions were disregarded. Only viscous and viscoelastic forces were considered. A uniaxial extension was also assumed as the radius variation in the axial direction is small. The Newtonian fluid has been considered prone to fluctuate during melt processing while the viscoelastic skin layer has a stabilizing effect. The velocity profile was affected by the choice of two characteristic parameters: extensional and shear-thinning parameters, as well as the viscoelastic skin flow rate ratio. Both shear-thinning and extensional parameters play important roles in melt flow. The results show that as the draw ratio increases, the system is dominated by the extensional parameters, whereas slow drawing is dominated by shearing, in spite of the thin viscoelastic skin.     

Effect of wall slip on the uniformity of flow from sheeting extrusion dies

A theoretical study for analyzing the uniformity of flow from sheeting extrusion dies is presented. In this study it is assume that a slip condition exists at the wall of the die, the magnitude of slip velocity is proportional to the shear stress at the wall, the flow is isothermal and steady state, and a power law model is valid for viscosity. Two extrusion dies, T-dies and coat-hanger dies, are examined. The flow uniformity at the exit of the die is calculated and compared with that for a nonslip analysis. The discrepancies between the slip and nonslip models imply that the wall slip condition induces a significant nonuniform flow distribution. Traditional design criticism based on the nonslip model are invalid for flow with the wall slip condition, and it is necessary to increase the length of the die land to even the flow distribution at the exit of the die.     

Two directional coextrusion flow of two molten polymers in flat dies

A theoretical study has been carried out of the two directional isothermal flow of two molten polymers in a flat die. The viscous behavior of each fluid is described by a power-law. Using an iterative finite difference method, a numerical program predicts the influence of geometrical and Theological parameters on the variations of the interface position. Theoretical values are in good agreement with experimental results obtained on a fish-tail die. In the case of a coat hanger-die, the mean value of the interface position is correct, but the general shape may be quite different.     

An experimental study of slip flow in capillaries and semihyperbolically converging dies

Slip velocity is measured in straight‐walled capillary dies and semihyperbolically converging dies (SHCDs) in three industrial polymer melts, with and without the presence of a viscosity reducing stearic acid (SA) additive. Data taken in shear flow through capillary dies indicated a substantial increase in the slip velocity caused by the addition of the SA. Data taken in the SHCDs showed much less of an increase in the slip velocity of the polymer/additive system relevant to the neat polymer. One possible explanation of this observation is that the magnitude of the slip velocity is directly related to the degree of orientation within the flowing polymeric material. POLYM. ENG. SCI., 47:159–167, 2007. © 2007 Society of Plastics Engineers     

Single screw extrusion of viscoplastic fluids subject to different slip coefficients at screw and barrel surfaces

Commonly encountered viscoplastic fluids including concentrated suspensions of polymeric and ceramic composites, foams, gels, concrete, food products, and energetic compounds exhibit wall slip during their flow and processing. For some viscoplastics fluids, especially highly filled suspensions, wall slip may dominate the flow and deformation and hence the processing behavior of the suspension. The wall slip velocity is generally a function of the wall shear stress and temperature. Various factors including the materials of construction i.e., chemical nature and the roughness of the wall surface affect the wall slip behavior of viscoplastic fluids. In this study an analytical model of the extrusion of viscoplastic fluids under isothermal and fully-developed conditions in shallow channels is developed. The model accommodates the use of different slip coefficients at barrel and screw surfaces. It thus permits the investigation of effects of introducing different materials of construction for the barrel and screw surfaces and development of design expressions.     

Gross melt fracture mitigation in converging dies: A singular behavior due to polymer wall slip

This work focuses on mitigating the gross melt fracture defect of polymer flowing through axisymmetrical and two‐dimensional dies. The die entrance angle is considered as well as the influence of the converging wall roughness. Singular results are obtained with a random styrene butadiene rubber (SBR) copolymer, as the gross melt fracture defect cannot be eliminated or mitigated by reducing the die entrance angle. Other experiments carried out with rough converging dies do not give better results. Indeed, the polymer essentially slips along the walls, as shown from capillary rheometer and birefringence experiments. Thus, these results point out the importance of elongational stresses and interfacial conditions in the die entrance region on flow instabilities and the gross melt fracture defect.     

An analytical model for material line kinematics in steady and unsteady buoyancy driven flows in 2-d channels

In this note, we use both developing and fully developed buoyant flow in a 2-d channel as a simple flow model useful for approximating buoyancy driven interpenetrating mixing phenomenon, such as the Rayleigh-Taylor instability. Our approach towards understanding and quantifying mixing within this flow follows the development by Ottino (The kinematics of mixing: stretching, chaos and transport, Cambridge University Press, Cambridge, 1989) that mixing is the kinematically efficient stretching and folding of material lines. Like Ottino, we use simple flows as a prototype to understand more complex problems. Here we derive the stretch length i.e. the deformation of a unit filament subjected to both the fully developed and developing buoyant channel flow. We use the simpler fully developed flow to compute the specific rate of stretching for the fully developed flow. An early time contraction-expansion effect is noted for stretching. Asymptotically, the stretching rate decays by the expected t⁻¹ behavior. This note indicates that early time buoyancy induced inter-penetrative mixing is a typical 2-d, shear driven phenomenon, exhibiting neither strong re-orientation (with attendant enhance mixing) nor negligible filament deformation (indicating minimal disruption of material surfaces).     

An analytical model for the prediction of power consumption for shear-thinning fluids with helical ribbon and helical screw ribbon impellers

An approximate analytical model has been developed to predict power consumption for the mixing of shear-thinning fluids with helical ribbon and helical screw ribbon impellers in the laminar flow regime. Extensive data on power input measurements embracing a wide range of flow behaviour index, with strong (n<0.4) and weak (0.4<n<1) shear-thinning fluid characteristics, available in the literature have been used to demonstrate the applicability of the present model for a wide range of helical ribbon mixer configurations. The model is able to explain the differences in the data reported in the existing literature and to successfully predict the complex dependence of power consumption on the fluid properties and the system geometry. Finally, the proposed correlation only requires a knowledge of the flow behaviour index of the fluid and of the geometrical parameters of the mixing systems (wall clearance, number of ribbons, pitch and width of the ribbons) and one characteristic parameter K_p of the mixing system which can be obtained from a single measurement of power for Newtonian liquids in the laminar regime.     

Numerical simulation of entry and exit flows in slit dies

A general-purpose finite element program has been used to simulate the flow of Newtonian, power-law, and viscoelastic fluids in the entry and exit regions of a slit die. It was found that shear-thinning increases the entrance correction while it decreases the exit correction. Shear-thinning reduces the size of the small corner vortex that forms in the entry flow of a Newtonian fluid. The swelling ratio had a value of 1.196 for Newtonian fluids and decreased as the value of the power-law index decreased. Viscoelastic calculations were performed using the Criminale-Ericksen-Filbey (CEF) constitutive equation. Convergence of the iterative scheme was unattainable for Deborah numbers above 1.0. The results showed a decrease of the entrance correction and an increase of the exit correction with elasticity. Extrudate swell first decreased slightly and further increased with the Deborah number.     

Numerical flow simulation of viscoplastic fluids in annuli

This paper describes numerical solutions for the laminar flow of non-Newtonian fluids in vertical annuli using the Herschel-Bulkley model to describe the rheological behaviour of such materials. Numerical solutions have been obtained when there is both axial and tangential flows in either a concentric or eccentric annulus. The tangential flow arises from the rotation of the inner cylinder of the annulus and the axial flow from a constant axial pressure gradient. The flow is analysed by solving the momentum and continuity equation numerically using the finite element method. The dimensionless velocity, deformation and stress profiles with other quantities such as the apparent viscosity and pressure distribution have been calculated for various eccentricities, radius ratios, fluid properties and flow parameters; the results give insights into the flow behaviour in the annuli. It is shown that the inclusion of rotational effects, for a fixed pressure gradient, is likely to increase the axial volumetric flowrate over non-rotating situations in concentric geometries. New results reveal that, in eccentric annuli, the situation is reversed and the flowrate gradually decreases as the rotation rate is increased.     

Temperature fields in extruder dies with circular,annular, or slit cross-section

Viscous dissipation causes significant temperature increases in polymers while they are flowing through extruder dies. The development of the temperature field has been studied numerically using the dimensionless parameters Na and Gz. With the annular geometry (parameter: ratio of radii κ) the well known developing temperature fields in capillaries could be compared with the developing temperature fields in an annulus and in a plane slit. The shear dependence of the viscosity is described by a “power law” and the temperature dependence by an exponential function. A simple graphical method is suggested for estimating temperature fields in extruder dies, making use of the fully-developed temperature field for very long dies as a reference state. For the demonstration of results, a power law exponent n = 0.4 has been chosen.     

An approximate analytical model for electrolyzers with axial dispersion

The steady state and transient behaviour of electrolyzers with axial dispersion is described in terms of an approximate mathematical model.     

Inertia and gravitational effects in extrusion dies for non-Newtonian fluids

An analysis is presented which allows the sheet or film die designer to estimate when inertial and gravitational effects are important. General theoretical equations are developed for end fed dies with arbitrary variation of the cavity cross sectional shape, cavity taper, slot length, and gap over the width. The method assumes viscous flow and a two dimensional approximation for the cavity flow. For fluid flow properties, it is assumed only that the apparent viscosity is a single valued function of the shear rate. In the important special case of constant die geometry and power law fluids, three dimensionless numbers plus the power law index are the parameters controlling the uniformity of flow from the die. Results are presented that illustrate when die orientations with respect to gravity and when fluid inertia are important. When they are not, simple expressions for die inlet pressure and uniformity index are given.     

Energy model of the interfacial slip of polymer blends under steady shear

Interfacial slip at high‐density polyethylene (HDPE)/polystyrene (PS) and high‐impact polystyrene (HIPS)/PS interfaces under steady shear was studied. The multilayer structure and energy model for steady shear proposed by Lam and colleagues was employed. Results indicated that there was no interfacial slip at the HIPS/PS interface. However, interfacial slip was detected for HDPE/PS under steady shear. Small interfacial thickness and weak interactions between HDPE and PS was proposed as the reason for interfacial slip at the HDPE/PS interface. Chain orientation under shear was believed to promote chain disentanglement in the interfacial layer and therefore increase interfacial slip. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 89: 1164–1470, 2003     

Theoretical prediction of the pressure gradients in coextrusion of non-newtonian fluids

In previous publications we have reported our experimental observations that, when the less viscous component flows outside the more viscous component in sheath-core coextrusion through a cylindrical die, or in sandwich three layer film coextrusion through a slit die, the pressure gradients in a two-phase system can be lower than those in the two components flowing individually. The experimental observations suggest that the energy requirement for extrusion can be decreased, and the throughput increased, when one judiciously chooses proper combinations of materials and/or optimal processing conditions. In this paper, we present a theoretical analysis, which shows that the seemingly anomalous behavior of the experimentally observed pressure gradient reduction is indeed possible, depending on the rheological properties of the individual components involved in coextrusion.     

Shear rate and wall slip velocity functions of polyvinyl chloride melts based on slit die viscometry data

Shear stress versus shear rate and wall shear stress versus slip velocity relations of polymer melts are key material property functions needed in the design of polymer processing equipment. These properties are normally measured using capillary or slit viscometers. The presence of slip greatly complicates the processing of the resulting viscometry data. This paper describes the conversion of the viscometry data of a number of polyvinyl chloride melts through slit dies with different gaps into these two material property functions. The conversion procedure, based on Tikhonov regularization, has the advantage that it is independent of rheological constitutive equation and does not require extrapolation of the experimental data. It also has the ability to cope with the unavoidable noise in the experimental data. Consequently, the property functions thus obtained are likely to be closer to true material properties than those from some of the existing methods. For each of the polyvinyl chloride melts investigated, these properties' functions will be compared with published results and their differences discussed.     

The flow distribution of molten polymers in slit dies and coathanger dies through three-dimensional finite element analysis

The isothermal flow of power-law fluids in slit dies and coathanger dies is studied. A general three-dimensional finite element code is developed for the purpose of flow analysis. The pressure distribution, the velocity distribution, and the transverse flow rate distribution are obtained. The effect of the die geometry on the flow distribution is critically discussed. It is found that a die channel with cross section of dog bone profile produces a flatter transverse flow rate distribution.